Stochastic Differential Equations

A rigorous geometric web-book for graduate students in mathematical finance who have already completed a serious probability course.

This site is designed to be read like an online book: precise definitions, theorem/proof structure, geometric representations, computational visuals, and finance-facing interpretation.

Philosophy

This course treats stochastic calculus as a mathematical language for describing random motion in continuous time. The objective is not merely to memorize formulas such as

$$(dW_t)^2 = dt,$$

but to understand why this symbolic rule is forced by the geometry of Brownian paths.

Primary references

This web-book is written as an original exposition, but its mathematical backbone follows standard references:

• Bernt Øksendal, Stochastic Differential Equations: An Introduction with Applications.
• Steven Shreve, Stochastic Calculus for Finance II: Continuous-Time Models.
• Ioannis Karatzas and Steven Shreve, Brownian Motion and Stochastic Calculus.
• David Williams, Probability with Martingales.
• Daniel Revuz and Marc Yor, Continuous Martingales and Brownian Motion.

Exercises are written in the style of these texts and point toward their canonical problem themes: Gaussian processes, martingales, quadratic variation, Itô integration, diffusion models, and arbitrage-free pricing.

Reading path

1. From random walks to Brownian motion.
2. Construction of the Itô integral.
3. Itô's formula as geometry of second-order variation.
4. Existence and uniqueness for SDEs.
5. Classical diffusions: GBM, OU, CIR.
6. Martingales, equivalent measures, and Girsanov.
7. Black-Scholes, Feynman-Kac, and term-structure models.
8. Commodity and energy applications.